Let's say I have the following linear system: \begin{equation} \left[ \begin{array}{cccc} S&&L^{T}&&A^{T}&&0\\ L&&0&&0&&0\\ A&&0&&0&&B\\ 0&&0&&B^{T}&&0 \end{array} \right] \cdot \left[ \begin{array}{c} u\\ p\\ \lambda \\ u_{p} \end{array} \right] = \left[ \begin{array}{c} f\\ 0\\ 0 \\ 0 \end{array} \right] \end{equation}
Let's say I solve it by using a LU decomposition, it is well known that if I have the same system but with a different RHS ($f_{1}$ instead of $f$) I can solve the new system using the same LU decomposition making the solution very fast.
My question is, what if I have a smaller version of the system, for example: \begin{equation} \left[ \begin{array}{cc} S&&L^{T}\\ L&&0 \end{array} \right] \cdot \left[ \begin{array}{c} u\\ p \end{array} \right] = \left[ \begin{array}{c} f\\ 0\\ \end{array} \right] \end{equation}
that is exactly the same system eliminating $\lambda$ and $u_{p}$ from the unknowns, Can I relate the LU decomposition of this new system, to the LU decomposition of the bigger system so that the solution of this sub-problem can be fast?
In practice what I would like to do is to avoid solving a big system and then again a smaller version of it, but i'm looking for a shortcut that speeds up the computation.